Radial Basis Function Artificial Neural-Network-Inspired Numerical Solver

A framework for a mesh-free numerical solver of differential equations is presented in this paper. Development of the solver is derived from machine learning techniques using artificial neural networks with Gaussian radial basis functions for their neurons. The proposed method incrementally develops an approximation through the optimization of a scalar condensed form of the differential equations. Unlike traditional solvers that require grids, volumes, or meshes, along with corresponding connectivity data, the proposed framework requires only a list of independent variable values to approximate the solution. Because of this, there is no need for the derivation or inversion of system matrices. Results are presented demonstrating the stability and accuracy of the proposed method and it is demonstrated that the spatial error estimate can exceed that of traditional methods.